Application of Level Set Method for Simulating Kelvin-Helmholtz Instability Including Viscous and Surface Tension Effects

Transcription

1 International Journal of Mathematis and Comutational Siene Vol. 1, o. 2, 2015, htt:// Aliation of Level Set Method for Simulating Kelvin-Helmholtz Instability Inluding Visous and Surfae Tension Effets Ashraf Balabel * Mehanial Engineering Det., Taif University, Taif, Hawiyya, Kingdom of Saudi Arabia Abstrat In the resent aer, the hydrodynami instability nown as Kelvin-Helmholtz instability is numerially investigated with the aliation of the level set method. First, the lassial Kelvin-Helmholtz instability, whih is onerned with the toologial hanges of interfae searating two immisible, inomressible and invisid fluids (normally liquid/gas) in irrotational and relative motion, is redited. Further, as in real flow, the effets of visosity and surfae tension are inluded. The numerial strategy is based on the solution of avier-stoes equations over a regular and strutured two-dimensional omutational grid using the ontrol volume aroah in both hases with a searate manner. The inemati and the dynami boundary onditions between the two hases are alied on the interfae. The transient evolution of the interfae due to the different surfae fores is redited by the level set method. The obtained results showed that the roll-u mehanism starting at the interfae is largely affeted by the different flow regimes onsidered. The inlusion of the visous fore an lead to a weaened roll-u mehanism. Moreover, the finite surfae tension fore an revent the roll-u mehanism omared with lassial Kelvin- Helmholtz instability. Keywords Kelvin-Helmholtz Instability, Level Set Method, umerial Simulation, Surfae Tension, Two-Phase Flow Reeived: Marh 7, 2015 / Aeted: Marh 20, 2015 / Published online: Marh 23, 2015 The Authors. Published by Amerian Institute of Siene. This Oen Aess artile is under the CC BY-C liense. htt://reativeommons.org/lienses/by-n/4.0/ 1. Introdution Many of industrial and engineering aliations enounter a large number of two hase flows; e.g. atomization of liquid jets [1], drolet dynamis [2], bubble flow [3], and other multihase flow systems [4]. The nonlinear stability of an interfae between two inomressible, invisid and irrotational fluids ontaining a veloity disontinuity remains a onsiderable hallenge to mathematiians and numerial analysts. Helmholtz [5] was the first to remar on the instability of those interfaes. Kelvin [6] has disussed the onditions under whih a lane surfae of water beomes unstable. As a result of the veloity disontinuity resented initially in the flow, a vortex flow is rodued and the motion of the interfae is subjeted to a well-nown Kelvin-Helmholtz instability (KHI). The basi mehanism of KHI develoment is in the existene of a uniform veloity shear between two different fluids. In most ases, the surfae tension and visosity are negleted in order to simlify the theoretial and numerial alulations and, onsequently, KHI is onsidered as a fluid instability that a lighter fluid is suerosed on a heavier one. The inluding of the gravity fore or the density stratifiation, without any uniform shear onsidered, leads to the well nown ases of Rayleigh-Taylor Instability or Rihtmyer- Meshov Instability. * Corresonding author address: ashrafbalabel@yahoo.om

2 International Journal of Mathematis and Comutational Siene Vol. 1, o. 2, 2015, The attention of KHI has reeived is due to in art to their imortane as models of free shear layers in high Reynolds number flows. This, in turn, determines the rate of mixing in the flow. On the other hand, and based on the exerimental observation, the interfaial roll-u is a relude to dro formation in free surfae flow evolution. The motion of the interfaes subjeted to the vortex flows is widely investigated by means of different numerial methods; see for examle [7, 8]. In real systems, the interfae is a layer with a finite thiness; in addition, visosity or surfae tension affets the interfae evolution. The effet of visosity on the evolution of KHI an be seen in suressing the short-wave disturbane and in thiening the vortex sheet. It means that the roll-u was weaened by visosity effets and the visosity fore ats as a regularization effet on the roll-u of vortex sheets. The singularity formation is one of the most interesting henomenons in the simulations of free surfae flows and it is onsidered disastrous from the stand oint of the numerial omutations. In the absene of the surfae tension, an aearane of a saw-tooth-lie instability in the interfae rofiles has been observed. This instability revents the simulations from roeeding to the late time highly nonlinear hase of the motion as singularities an form on the interfae in finite time [7]. However, the inlusion of the surfae tension effet was found to revent the aearane of the numerial instability but only for finite times. Thereafter, loally irregular behavior eventually reaeared and resisted subsequent attemts at numerial smoothing. These irregularities are develoed on the interfae rofile where some form of oherent roll-u might be exeted. Although several numerial and/or hysial mehanisms are disussed that might rodue irregular behavior of the disretized interfae in the resene of the surfae tension term, the basi ause of this instability remains unnown. Aording to [8], there is no ublished analyti study that has demonstrated that the surfae tension effet removes the illosedness of the vortex sheet motion in both the linear theory and the nonlinear ase. Furthermore, a saw-tooth-lie instability that aeared in the solution, generally growing rather more raidly in ases involving finite surfae tension term. After greater values of time, the interfae did not roll u smoothly but rather develoed irregularities on the interfae. Aording to the exerimental omlexity of suh roblem, exat measurements of the interfae shaes are not available; however, the theoretial as well as the numerial treatments of the Kelvin-Helmholtz instability have enountered some onstraints related to the nature of the roblem as it inludes a transient, non-uniform free surfae flow with large satial and temoral gradients. As a result of the huge develoment in the numerial simulation of two-hase flows, arefully exeuted simulations in suh ontext an virtually relae exeriments [9]. In general, the numerial reditions of two-hase flow dynamis have been limited in auray artly by the erformane of three ey elements, viz.: develoment of the omutational algorithm, interfae traing methods, and turbulene redition models [10]. A variety of numerial methods have been reently develoed and validated to two-hase flow. However, an effiient and omlete numerial method is not available. An extended review of numerial tehniques alied for twohase flow inluding adv/disadvantages an be found in [11]. More reently, the author has develoed a new numerial method, nown as Interfaial Marer Level Set Method (IMLS), for rediting the omlete dynamis of two-hase flows. This numerial method is validated and its auray is estimated through the erforming of a wide range of industrial and engineering aliations [12-18]. This method is also further alied in the resent aer and shortly exlained in the following setions. 2. Comutational Domain Consider the two-dimensional motion of two immisible, inomressible, invisid fluids of equal density searated by an interfae. The retangular oordinates (x,y) have been hosen, with the x-axis horizontal and aligned with the initial mean level of the interfae, and the y-axis vertial to it. The interfaial shae is assumed to be eriodi in the x-diretion with wavelength. The motion of the two fluids an be desribed in terms of the motion of a single wavelength of the interfae. The fluid above the x-axis moves at a uniform veloity U, and below at uniform veloity of equal magnitude but in the oosite diretion, as shown in Fig.(1). Figure 1. Configuration of the surfae disontinuity At time=0, the surfae onfiguration an be onsidered as:

3 32 Ashraf Balabel: Aliation of Level Set Method for Simulating Kelvin-Helmholtz Instability Inluding Visous and Surfae Tension Effets 2πx ys = asin λ where a is the disturbane amlitude. 3. Mathematial Formulations The governing equations for inomressible two-hase flow are mathematially exressed by the onservation equations of the mass and momentum at eah oint of the flow field. These equations an be written in the rimitive variables formulation in form of ontinuity equation and avier-stoes equations, resetively, as follows: u =0 (1) i u i ρ i + ( uu i i ) = 2 i + µ i u (2) i t where ρ, u,, µ are the density, veloity vetor, ressure and visosity of the fluid. The subsrit i denotes the liquid hase (i=1) and the gas hase (i=2), resetively. It is nown that, for two hase flows, the ressure and veloity field are aused by the external gas stresses in the tangential and normal diretions. Therefore, the dynami balane between both hases an be obtained by solving the omlete set of the governing equations in the gas hase for the veloity and ressure for the gas hase surfae and using these values as boundary onditions for solving the interior of the liquid hase to seify the exat level of ressure and veloity omonents inside it. The obtained normal veloity omonent at the liquid interfae is then used to obtain the toologial hanges and deformation of the drolet surfae via the level set method. The fat that there is no ressure transort equation neessitates the onsideration of the ontinuity equation as a means to obtain the orret ressure field. This is done by a roer ouling between the ressure and veloity field through the Poisson equation for ressure: 2 ui i = ρ i + ( uiui ) (3) t The Poisson equation is solved by means of the Suessive Over-Relaxation method in eah hase to obtain the ressure level inside the onsidered hase. In our algorithm, the imliit frational ste-non iterative method is alied to obtain the veloity and ressure filed by resuming that the veloity field reahes its final value in two stages; that means n u + 1 = u * + u (4) * whereby, u is an imerfet veloity field based on a guessed ressure field, and u is the orresonding veloity orretion. Firstly, the 'starred' veloity will result from the solution of the momentum equations. The seond stage is the solution of Poisson equation for the ressure: 2 ρ = u t where will be alled the ressure orretion. One this equation is solved, one gets the aroriate ressure orretion, and onsequently, the veloity orretion is obtained aording to: u t = ρ This frational ste method desribed above ensures the roer veloity-ressure ouling for inomressible flow fields Interfaial Boundary Conditions The boundary onditions at the interfae, or jum onditions, are omrised of the dynami and inemati onditions. In the ase of two immisible fluids, referring to Fig.2, the dynami stress balane at the liquid interfae Γ may be written in the following general form: Figure 2. ormal and tangential veloity omonents of an arbitrary interfae * ( ) t (5) (6) [-I 2µ D] n = σκn + σ (7) where I is the identity matrix, is the ressure, µ is the dynami visosity, σ is the surfae tension oeffiient, and D is the rate of deformation tensor. The braet means the jum of the stresses along the fluid interfae Γ, the unit normal vetor n is taen from fluid 2 to fluid 1, t denotes the gradient in the loal free surfae oordinates, and t is an arbitrary vetor erendiular to the normal of the interfae. The urvature of the interfae κ is omuted from the

4 International Journal of Mathematis and Comutational Siene Vol. 1, o. 2, 2015, following equation: κ = n (8) The normal and tangential veloity omonents are V n and V t resetively. It is lear that, the effet of the surfae tension is to balane the jum of the normal stress along the fluid interfae. The seond term on the right hand side of the dynami stress equation, Eq.(7), is the stress due to the gradient on surfae tension in the loal interfae oordinates or the so-alled Marangoni effet, usually imortant when a temerature gradient is alied arallel to the interfae, e. g. thermoaillary onvetion. In ontrast to the revious two-hase numerial methods, in whih the interfaial jum onditions are embedded naturally on the momentum equations by alying CSF model [19], the jum onditions between the two fluids are treated here as a boundary onditions enfored exliitly at the interfae Γ. Taing the rojetions of the jum onditions in the diretions normal and tangential to the interfae, onsidering a onstant surfae tension, one obtains the following two equations in the normal and tangential diretions, resetively: [ 2µ ( u n) n] = σκ (9) [ ( u n) t + µ ( u t) n] = 0 µ (10) It is notied from the above equations that surfae tension effets are inluded in the normal stress balane, while the equality of the shear stress is satisfied in the tangential diretion. In the seial ase of invisid flow with onstant surfae tension oeffiient the above equations are redued to the Young-Lalae equation whih determines the ressure jum aross the interfae. In addition to the equality of the dynami interfaial stresses desribed above, the inemati onditions should also be onsidered. When there is no mass transfer through the interfae, the inemati onditions is satisfied by assuming the ontinuity of the normal veloity omonents, i.e. interfae by solving the aroriate equation of the level set funtion Level Set Funtion The level set method is a tye of aturing methods where a defined hase funtion φ, is smoothed over the entire omutational domain. The level set funtion at any given oint is taen as the signed normal distane from the interfae with ositive on the liquid hase (i.e. φ>0), and negative on the gas hase (i.e. φ<0). Consequently, the interfae is imliitly defined as the zero level set of the level set funtion. The transort equation of the level set funtion an be desribed by the following equation: φ + u φ = 0 t (13) where u is the veloity vetor. The geometrial roerties of the interfae, (normal vetors and urvature), an be defined as: ϕ n =, κ = n ϕ (14) The original wor of level set method in two-hase numerial simulation is referred to [20]. A review of suh wors an be found in the ited review [16] Comutational Methodology In order to solve the above system of equations, the resribed ontrol volume formulation resented by [21] has been imlemented. However, aording to the omlexities introdued by the staggered grid system in two-hase flow alulations and the need to tra the liquid surfae, a nonstaggered grid system is alied here, whih requires the use of a single ell networ for all variables and a olloated seifiation of variables at the entre of eah ell. V ) = V ) (11) n l In addition to that, the liquid gas interfae requires equality of dynami stresses, eseially shear stresses, on both sides of the interfae, i.e. n g τ l = τ g (12) The system of the above equations and the resribed boundary onditions should be solved simultaneously to determine the flow field in the two fluids. The normal veloity omonents are then used for the advetion of the Figure 3. Calulation stenil for obtaining the intermediate flux veloity at ell faes

5 34 Ashraf Balabel: Aliation of Level Set Method for Simulating Kelvin-Helmholtz Instability Inluding Visous and Surfae Tension Effets The olloated seifiations of flow variables require a satial tehnique to revent ressure deouling between adjaent ells. The only drawba of the non-staggered grid system is that it requires a high-order aroximation for alulation of the fluxes and to obtain differene equations that an revent ressure osillations. In the resent study, the derivatives at the entral oint of the ontrol volume are aroximated using the seond-order entral differene sheme to ahieve the best auray. The veloity omonents at the ontrol volume faes are alulated using the average values of the two-edge oints that are alulated through the average of the four neighbouring grid oints, as seen in figure Initial Comlex Veloity Field In order to start the alulation, a veloity field has to be resribed. The initial veloity field lays a signifiant role in the roerly started alulations, and onsequently in the hysial deformation of the interfae. In KHI, a omlex veloity field an be obtained through a distribution of vortex-lie singularities on the interfae [22]. The veloity omonents indued at oint loated at the entre of the omutational domain. U u = v U = sinh[2π ( y y osh[2λ( y y os[2π ( x osh[2λ( y y os[2π( x sin[2π ( x x x x where is the number of the vorties on the interfae. In the resent wor only one vortex ole loated at the enter of the omutational domain. 4. Results and Disussion In the resent setion, three different ases with the data shown in Table 1 are erformed. The first ase is erformed for zero surfae tension and zero visosity fluids, while the other ases are erformed by inluding the visosity and surfae tension resetively. Table 1. Data for the erformed test ases Proerty ρ 1/ ρ 2 µ 1/ µ 2 σ (/m)( Grid x, y t Case I x151 1e -4, 1e -4 3e -7 Case II x151 1e -4, 1e -4 3e -7 Case III x151 1e -4, 1e -4 3e Case I The evolution of an initial sinusoidal disturbane with amlitude a = 0.06λ is shown in Fig.(4). It is lear that the roer initial veloity distribution rodues a good behaved smooth roll-u and onentration of vortiity at the midoint of the wavelength (x =λ/2). o singularities an be observed. The roll-u roess agrees well with the revious exetations [22, 23], and the interfae forms at later times a smooth siral. In omarison with the results [24], the roosed numerial roedure an roerly redit the evolution of the interfae without any numerial roblems. That reveals the effetiveness of the numerial method adoted in order to obtain a real hysial henomenon. Figure 4. Evolution of KHI for Case I

6 International Journal of Mathematis and Comutational Siene Vol. 1, o. 2, 2015, Case II In order to show the effet of visous fore on the evolution of the KHI, the revious test ase (Case I) is erformed, however, the visosity term is inluded in the governing equations with a visosity ratio between the two fluids equal 1. The inlusion of the visosity effet in two-hase simulations is normally a hallenge tas due to the existene of the interfaial visous stresses and the diffiulties assoiated with its numerial modelling. From Fig. 5, it an be seen that the visosity ats to thien the vortex sheet and tries to suress the roll-u mehanism observed in the invisid simulations. The numerial results obtained for the visous simulation redit that visosity ats as a regularization effet on the roll-u of vortex sheets. These observations oinide with the revious investigations [24]. Figure 5. Evolution of KHI for Case II 4.3. Case III In the resent setion, the effet of the finite surfae tension on the rolling-u mehanism of the initial wave disturbane is resented, while the visous effets are negleted. By inluding the finite surfae tension into the numerial simulation, it is observed that the roll-u an our as shown in Fig. 6. However, it an be observed a henomenon alled "inhing singularity" in ontrast to the roll-u mehanism for ase I. Moreover, the rolling-u mehanism in ase of finite surfae tension is weaened in omarison with the results of Case I. Figure 6. Evolution of KHI for Case III 5. Conlusion The resent aer investigates numerially the nown Kelvin-Helmholtz instability between two fluids with different flow regimes. The numerial simulation is arried out by solving avier-stoes equations ouled with level set method in both fluids. The interfaial boundary onditions are well desribed in both fluids. The results indiate that, in ase of invisid simulation the roll-u mehanism of the interfae searating the two fluids is well redited. By inluding the visosity effet in the numerial simulation, the roll-u mehanism is weaened. The effet of inluding the surfae tension an be seen in observing the inhing singularity henomenon alled in ontrast to the roll-u mehanism for invisid flow. In general, the alied numerial method is seen to be effetive and robust in twohase flow simulations. Referenes [1] Lefebvre, A. H. (1989): Atomization and srays, Hemishere Publishing Cororation. [2] Fuster, D., Agbaglah, G., Josserand, C., Poinet, S. and Zalesi, S. (2009): umerial simulation of drolets, bubbles and waves: state of the art, Fluid Dyn. Res. 41, [3] Sato, Y. (1975): Liquid veloity distribution in two hase bubble flow, Int. J. Multihase flow, vol. 2(1), [4] Kolev,.I., (2007): Multihase flow dynamis: Thermal and Mehanial Interation", Sringer. [5] Helmholtz, H., Ueber Disontinuirlihe Fluessigeitsbewegungen. Monatberihte Aad. d. Wiss. Berlin, vol.23, , 1868.

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